Quelle schumu.xml Sprache: XML

<Chapter><Heading>Approximating the Schur multiplier</Heading>

The algorithm in <Cite Key="MR2678952"/> approximates the Schur multiplier of
an invariantly finitely L-presented group by the quotients in its
Dwyer-filtration. This is implemented in the &lpres;-package and the
following methods are available:

<Section><Heading>Methods</Heading>

<ManSection><Oper Name="GeneratingSetOfMultiplier" Arg="lpgroup"/>
<Description>
uses Tietze transformations for computing an equivalent set of
relators for <A>lpgroup</A> so that a generating set for its Schur
multiplier can be read off easily.
</Description>
</ManSection>

<ManSection><Oper Name="FiniteRankSchurMultiplier" Arg="lpgroup c"/>
<Description>
computes a finitely generated quotient of the Schur multiplier of
<A>lpgroup</A>. The method computes the image of the Schur multiplier
of <A>lpgroup</A> in the Schur multiplier of its class-<A>c</A>
quotient.
</Description>
</ManSection>

<ManSection><Oper Name="EndomorphismsOfFRSchurMultiplier" Arg="lpgroup c"/>
<Description>
computes a list of endomorphisms of the `FiniteRankSchurMultiplier' of
<A>lpgroup</A>. These are the endomorphisms of the invariant
L-presentation induced to `FiniteRankSchurMultiplier'.
</Description>
</ManSection>

<ManSection><Oper Name="EpimorphismCoveringGroups" Arg="lpgroup d c"/>
<Description>
computes an epimorphism of the covering group of the class-<A>d</A>
quotient onto the covering group of the class-<A>c</A> quotient.
</Description>
</ManSection>

<ManSection><Oper Name="EpimorphismFiniteRankSchurMultiplier" Arg="lpgroup d c"/>
<Description>
computes an epimorphism of the <M>d</M>-th `FiniteRankSchurMultiplier'
of the invariant <A>lpgroup</A> onto the <M>c</M>-th
`FiniteRankSchurMultiplier'. Its restricts the epimorphism
`EpimorphismCoveringGroups' to the corresponding finite rank
multipliers.
</Description>
</ManSection>

<ManSection><Func Name="ImageInFiniteRankSchurMultiplier" Arg="lpgroup c elm"/>
<Description>
computes the image of the free group element <A>elm</A> in the
<A>c</A>-th `FiniteRankSchurMultiplier'. Note that elm must be
a relator contained in the Schur multiplier of <A>lpgroup</A>;
otherwise, the function fails in computing the image.<P/>

The following example tackels the Schur multiplier of the Grigorchuk
group.
<Example><![CDATA[
gap> G := ExamplesOfLPresentations( 1 );;
gap> gens := GeneratingSetOfMultiplier( G );
rec( FixedGens := [ b^-2*c^-2*d^-2*b*c*d*b*c*d ],
  IteratedGens := [ d^-1*a^-1*d^-1*a*d*a^-1*d*a,
      d^-1*a^-1*c^-1*a^-1*c^-1*a^-1*d^-1*a*c*a*c*a*d*a^-1*c^-1*a^-1*c^-1*a^
        -1*d*a*c*a*c*a ],
  BasisGens := [ a^2, b*c*d, b^-2*d^-2*b*c*d*b*c*d, b^-2*c^-2*b*c*d*b*c*d ],
  Endomorphisms := [ [ a, b, c, d ] -> [ a^-1*c*a, d, b, c ] ] )
gap> H := FiniteRankSchurMultiplier( G, 5 );
Pcp-group with orders [ 2, 2, 2 ]
gap> GeneratorsOfGroup( H );
[ g15, g17, g16 ]
gap> EndomorphismsOfFRSchurMultiplier( G, 5 );
[ [ g15, g16, g17 ] -> [ g15, id, g16 ] ]
gap> Kernel( last[1] );
Pcp-group with orders [ 2 ]
gap> GeneratorsOfGroup( last );
[ g16 ]
gap> EpimorphismFiniteRankSchurMultipliers( G, 5, 2 );
[ g15, g16, g17 ] -> [ g10, id, g13 ]
gap> Range( last ) = FiniteRankSchurMultiplier( G, 2 );
true
gap> Kernel( EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ) );
Pcp-group with orders [ 2 ]
gap> GeneratorsOfGroup( last );
[ g16 ]
gap> Kernel( EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ) ) =
</A> Kernel( EndomorphismsOfFRSchurMultiplier( G, 5 )[1] );
true
gap> ImageInFiniteRankSchurMultiplier( G, 5, gens.FixedGens[1] );
g15
gap> ImageInFiniteRankSchurMultiplier(G,5,Image(gens.Endomorphisms[1],
</A> gens.IteratedGens[1] ) );
g16
gap> ImageInFiniteRankSchurMultiplier(G,5,gens.IteratedGens[1] );
g17
]]></Example>
</Description>
</ManSection>

</Section>

</Chapter>

quality92%

¤ Dauer der Verarbeitung: 0.12 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.